Numerical Method Based on DGM for Solving the System of Equations Describing Motion of Viscoelastic Fluid with Memory

A Stable and Convergent Finite Difference Scheme for 2D Incompressible Nonlinear Viscoelastic Fluid Dynamics Problem

“…This FD scheme is first established because at present there has not been any FD scheme reported for the 2D incompressible nonlinear viscoelastic fluid dynamics problem. This implies that the technique here is development for the existing results (see, e.g., [7][8][9][10][11][12][13]).…”

“…Then the FD solutions Proof. By using Taylor's formula to expand (10) and (13) -(16) at the reference point 5 , 6 , 7 #, (11) at the reference point 5"!/ , 6 , 7 # , (12) at the reference point 5 , 6"!/ , 7 #, or from the discretizing procedure of (10) -(16), we could obtain the following error estimates (30). This completes the proof of Theorem 2.…”

“…Faria and Karam-Filho have also proposed a regularized-stabilized MEF formulation for the steady flow of an incompressible fluid of Bingham type in Reference [13], but their problem is also stationary and linear. Whereas the system of Eqs (1) - (9) here is nonlinear, includes seven unknown functions, and the unknown deformation tensor may be an unsymmetrical matrix, therefore, it is thoughtfully different from those problems in References [7][8][9][10][11][12][13] and is more complex than those equations in References [7][8][9][10][11][12][13]. Thus, either the establishment of the numerical models or the theoretical analysis of the stability and convergence as well as the error estimates of the numerical solutions for the system of Eqs (1) -(9) includes more difficulties and requires more techniques than those in the above References [7][8][9][10][11][12][13], but it is more useful.…”

“…Though Barton and Kirby have established a finite difference (FD) scheme for fluid flow problems on nonstaggered grids in [7], Khader and Megahed have presented a numerical simulation using the FD method for the flow and heat transfer in [8], and Pandit et al also have built a transient higher order compact scheme for incompressible viscous flows on geometries beyond rectangular in [9], their problems are different from the system of Eqs (1) - (9). Although Soukup has presented a numerical method based on DGM for solving the system of equations describing motion of viscoelastic fluid with memory in Reference [10], the problem discussed is linear and a transformed Navier-Stokes equation, and the existence and uniqueness as well as the convergence (error estimates) of the numerical solutions had not been studied. Sampaio has also developed some mixed finite element (MEF) models for incompressible viscous flows in References [11,12], but has yet not provided the theoretical analysis of the existence and uniqueness as well as the convergence (error estimates) of the numerical solutions.…”